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On Interval Valued Generalized Difference Classes Defined by Orlicz Function

Ayhan Esi, Bipan Hazarika*

Department of Mathematics, Gandhi University, Rono Hills, Arunachal Pradesh, India

Abstract: In this paper, using the difference operator and Orlicz functions, we introduce and examine some generalized difference sequence spaces of interval numbers. We prove completeness properties of these spaces. Further, we investigate some inclusion relations related to these spaces.

Keywords: sequence space, interval numbers, difference sequence, completeness

Received October 14, 2013; Revised November 09, 2013; Accepted November 11, 2013

Citation: Ayhan Esi, Bipan Hazarika, On Interval Valued Generalized Difference Classes Defined by Orlicz Function. Turkish Journal of Analysis and Number Theory, 1 (1). 48-53. January 2013. doi:10.12691/tjant-1-1-10

1. Introduction

The work of interval arithmetic was originally introduced by Dwyer [3] in 1951. The development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by Moore [15] and Moore and Yang [16]. Furthermore, Moore and others [3]; [4]; [10] and [17] have developed applications to differential equations. Chiao in [2] introduced sequence of interval numbers and defined usual convergence of sequences of interval numbers. Şengönül and Eryilmaz in [20] introduced and studied bounded and convergent sequence spaces of interval numbers. Recently Esi studied strongly λ-and strongly almost λ-convergent sequences spaces of the interval numbers in [5], respectively. Also, Esi studied some new type sequence space of the interval numbers in [6,7] and lacunary sequence spaces for interval numbers in [8]. In Hazarika [11] introduced the notion of λ-ideal convergent interval valued di¤erence classes defined by Musielak-Orlicz function.

Kizmaz [12] introduced the notion of di¤erence sequence spaces as follows:

for and Later on, the notion was generalized by Et and Çolak [9] as follows:

for and where and also this generalized difference notion has the following binomial representation:

Recall in [18], [13] that an Orlicz function M is continuous, convex, non-decreasing function define for such that and for and as If convexity of Orlicz function is replaced by then this function is called the modulus function and characterized by Ruckle [19]. An Orlicz function is said to satisfy for all values ,if there exists such that Subsequently, the notion of Orlicz function was used to defined sequence spaces by Altin et. al., [1], Tripathy and Mahanta [21], Tripathy et. al., [22], Tripathy and Sarma [23] and many others.

2. Preliminaries

A set consisting of a closed interval of real numbers such that is called an interval number. A real interval can also be considered as a set. Thus we can investigate some properties of interval numbers, for instance arithmetic properties or analysis properties. We denote the set of all real valued closed intervals by Any elements of is called closed interval and denoted by That is An interval number is a closed subset of real numbers (see [2]). Let and be first and last points of the interval number respectively. For we have

and if , then

and if , then

The set of all interval numbers is a complete metric space under the mertic defined by

[15].

In the special case and we obtain usual metric of

Let us define transformation by Then is called sequence of interval numbers and is called term of the interval numbers sequence The set of all sequences of the interval numbers denoted by cf. [2].

A sequence of interval numbers is said to be convergent to the interval number if for each there exists a positive integer such that for all and we denote it by

Thus, and [2].

A sequence space is said to be solid (or normal) if whenever for all sequences of scalars with for all

A sequence space is said to be symmetric if implies where is a permutation of

A sequence space is said to be monotone if contains the canonical pre-images of all its step spaces.

Let and be a sequence space. A K-step set of is a class of sequences A canonical pre-image of a sequence is a sequence defined as follows:

A canonical pre-image of a step set is a set of canonical pre-images of all elements in , i.e. is in canonical pre-image if and only is canonical pre-image of some

A sequence space is said to be sequence algebra if whenever

A sequence space is said to be convergence free if whenever and whenever , where is the zero element.

Remark 2.1. A sequence space is solid implies is monotone.

3. Main Results

In this paper we introduce and examine some generalized difference sequences of interval numbers using the Orlicz functions.

Definition 3.1. Let be a sequence of interval numbers and be an Orlicz function. We define the following sequence spaces:

where

Throughout the paper, will denote any one of the notation and

Theorem 3.1. and are complete metric spaces with the metric

Proof. Let be any Cauchy sequence in where for each Then for given For a fixed and choose such that Then there exists such that

[picture] | (3.1)

Hence

Then is a Cauchy sequence in and so is a convergent sequence in Let Again from (3.1)

Hence is a Cauchy sequence in for all and so is a convergent sequence in for all Let for all

For we have

Similarly we have

Thus exists. Let Proceeding in this way inductively we conclude that for all Using continuity of , we have

Thus for all we obtain that

That is

Then the inequality

implies that This completes the proof.

Theorem 3.2. The classes of interval numbers of sequences and are nowhere dense subsets of

Proof. From Theorem 3.1. we have and are closed subsets of the complete metric space Also and are proper subsets which follows from the following example.

Example 3.1. Let and Consider the interval sequence defined as follows:

and

Thus but Hence the result.

Theorem 3.3. for and the inclusions are strict.

Proof. We give the proof for the inequality only. The rest of the results follows similar way. Let Then for some we have

[picture] | (3.2)

Since

Then from the equation (3.2) and the continuity of , the result follows from the following relation

This shows that

To show that the inclusions are strict, consider the following examples.

Example 3.2. Let and Consider the sequence of interval numbers defined by

i.e. as and as Thus but Hence the inclusion is strict.

Example 3.3. Let and Consider the sequence of interval numbers defined by

Then Thus Hence the inclusion is strict.

Theorem 3.4. Let and be two Orlicz functions. Then

(i)

(ii)

Proof. (i) We prove the result for and the rest of the cases will follow similarly. Let Then for we have

[picture] | (3.3)

Let and with such that for . We write

Then for

we have

where and denotes the integer part of Given by the definition of Orlicz function for we have

for and using (3.3).

Again for

we have

for and using (3.3).

Thus for we have

Hence Thus

(ii) It will follows from the following inequality

The proof of the following result is also routine work.

Theorem 3.5. Let and be two Orlicz functions satisfying If then where

Theorem 3.6. The classes of sequences of interval numbers and are not sequence algebra in general.

Proof. The result follows from the following example.

Example 3.4. Let and Consider the two sequences of interval numbers defined by

Therefore for all we have

Thus Now, we have

i.e. This completes the proof.

Theorem 3.7. The classes of interval numbers of sequences and are not convergence free.

Proof. Let and Consider the interval sequence defined as follows:

and

Hence as Thus Let defined as follows:

and

Thus Therefore the classes of interval numbers and are not convergence free.

Theorem 3.8. The classes of interval numbers and are neither monotone nor solid.

Proof. Let and Consider the interval sequence defined by:

and

i.e. as Thus

Let be a subset of and let be the canonical pre-image of the J-step set of defined as follows: is the canonical pre-image of implies

Now

and

Thus Therefore the classes of interval numbers and are not monotone. By the Remark 2.1, these spaces are not solid.

Now let’s define the sequence by

and , thus

Let be a subset of and let be the canonical pre-image of the J-step set of defined as follows: is the canonical pre-image of implies

Now

and

Therefore and is not monotone. By the Remark 2.1, this space is not solid.

Theorem 3.9. The classes of interval numbers and are not symmetric.

Proof. The result follows from the following example.

Example 3.5. Let and Consider the interval sequence defined by

and

Thus Let the sequence of interval numbers be a rearrangement of the sequence of interval numbers defined as follows:

i.e.

Then for all odd and satisfying we have

From the last two equation, it is clear that is unbounded, thus Therefore the class is not symmetric.

References

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[2]   | Kuo-Ping Chiao, Fundamental properties of interval vector max-norm, Tamsui Oxford Journal of Mathematics, 18(2)(2002), 219-233.
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